Transition of large R-charge operators on a conformal manifold

Abstract

We study the transition between phases at large R-charge on a conformal manifold. These phases are characterized by the behaviour of the lowest operator dimension ∆(QR) for fixed and large R-charge QR. We focus, as an example, on the D = 3, $$ \mathcal{N} $$ N = 2 Wess-Zumino model with cubic superpotential $$ W= XYZ+\frac{\tau }{6}\left({X}^3+{Y}^3+{Z}^3\right) $$ W = XYZ + τ 6 X 3 + Y 3 + Z 3 , and compute ∆(QR, τ) using the ϵ-expansion in three interesting limits. In two of these limits the (leading order) result turns out to be$$ \Delta \left({Q}_{R,\tau}\right)=\left\{\begin{array}{ll}\left(\mathrm{BPS}\;\mathrm{bound}\right)\left[1+O\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)\right],& {Q}_R\ll \left\{\frac{1}{\epsilon },\kern0.5em \frac{1}{\epsilon {\left|\tau \right|}^2}\right\}\\ {}\frac{9}{8}{\left(\frac{\epsilon {\left|\tau \right|}^2}{2+{\left|\tau \right|}^2}\right)}^{\frac{1}{D-1}}{Q}_R^{\frac{D}{D-1}}\left[1+O\left({\left(\epsilon {\left|\tau \right|}^2{Q}_R\right)}^{-\frac{2}{D-1}}\right)\right],& {Q}_R\gg \left\{\begin{array}{ll}\frac{1}{\epsilon },& \frac{1}{\epsilon {\left|\tau \right|}^2}\end{array}\right\}\end{array}\right. $$ Δ Q R , τ = BPS bound 1 + O ϵ τ 2 Q R , Q R ≪ 1 ϵ 1 ϵ τ 2 9 8 ϵ τ 2 2 + τ 2 1 D − 1 Q R D D − 1 1 + O ϵ τ 2 Q R − 2 D − 1 , Q R ≫ 1 ϵ , 1 ϵ τ 2 which leads us to the double-scaling parameter, ϵ|τ|2QR, which interpolates between the “near-BPS phase” (∆(Q) ∼ Q) and the “superfluid phase” (∆(Q) ∼ QD/(D−1)) at large R-charge. This smooth transition, happening near τ = 0, is a large-R-charge manifestation of the existence of a moduli space and an infinite chiral ring at τ = 0. We also argue that this behavior can be extended to three dimensions with minimal modifications, and so we conclude that ∆(QR, τ) experiences a smooth transition around QR ∼ 1/|τ|2. Additionally, we find a first-order phase transition for ∆(QR, τ) as a function of τ, as a consequence of the duality of the model. We also comment on the applicability of our result down to small R-charge.

Large R-charge EFT correlators in N=2 SQCD

“…It was observed in [4,5] that the MMP contribution q (mmp) n has a well-defined doublescaling limit in the sense of 9 [2,75]. (Also in the same sense of the double-scaling limits taken in several nonsupersymmetric examples recently [31,42,[45][46][47][48][49][50]83])…”

“…One can study two-point functions of powers O n ∆ of the generator, and use analytic methods to estimate the behavior of the two-point function when the total R-charge J = n∆ of the operator insertion O n ∆ is large. In [3][4][5] we used a supersymmetric version of the previously developed large-charge expansion 2 [28][29][30][31][32][33][34][35][36][37][38][39][40] (see [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60] for more recent developments) to determine the leading behavior of the correlation function at large n, and then used supersymmetric recursion relations [61][62][63] to derive the higher 1 n corrections from the leading terms. All this analysis uses only large J as a control parameter; the analysis does not require any weak-coupling limit of a marginal parameter of the CFT, or for that matter any marginal parameter at all.…”

Semiclassics of three-dimensional SCFTs from holography

“…In particular, the Betti operators (11.35) are the holographic EFT realizations of the baryonic operators (2.33) of the microscopic quiver gauge theory. 39 Note that this matching implies the following microscopic interpretation of the holographic EFT chiral superfields ρ a :…”

“…The idea of looking at sectors of operators with large quantum numbers was introduced in the seminal papers [33,34], which have inspired several subsequent developments. Here we will adopt the general philosophy of [1][2][3], in which information on the large R-charge sector is obtained from BPS semiclassical states of the EFTsee also [5,[35][36][37][38][39][40] for other works on the large charge sectors of supersymmetric models related to the EFT approach. We will show how to do so for general toric models, deriving some universal results.…”

Spontaneously broken boosts in CFTs

“…This means that the effective field theory description of (4.1) is inappropriate, and should be replaced by the approach [31][32][33][34][35][36]. In fact sometimes these two types of effective theories are connected [37]. In our analysis of the boost symmetry realization on the large charge states in free field theory, we will not use an effective theory approach, rather, we will pursue a more straightforward analysis of the correlation functions.…”